'The Four Colour Theorem' printed from http://nrich.maths.org/

The Four Colour Conjecture was first stated just over 150 years
ago, and finally proved conclusively in 1976. It is an outstanding
example of how old ideas combine with new discoveries and
techniques in different fields of mathematics to provide new
approaches to a problem. It is also an example of how an apparently
simple problem was thought to be 'solved' but then became more
complex, and it is the first spectacular example where a computer
was involved in proving a mathematical theorem.

1. In the Beginning

The conjecture that any map could be
coloured using only four colours first appeared in a letter from
Augustus De Morgan (1806-1871), first professor of mathematics at
the new University College London, to his friend William Rowan
Hamilton (1805-1865) the famous Irish mathematician in 1852. It had
been suggested to De Morgan by one of his students, Frederik
Guthrie, on behalf of his elder brother Francis (who later became
professor of mathematics at the University of Cape Town).

Augustus De Morgan (1807-1871) and
William Rowan Hamilton (1805-1865)

The problem, so simply described, but so tantalizingly difficult
to prove, caught the imagination of many mathematicians at the
time. In the late 1860s De Morgan even took the problem and his
proof to America where among others, Benjamin Peirce (1809-1880) a
famous mathematician and astronomer, became interested in it as a
way to develop his logical methods.

De Morgan used the fact that in a map with four regions, each
touching the other three, one of them is completely enclosed by the
others. Since he could not find a way of proving this, he used it
as an axiom , the basis of
his proof.

A copy of De Morgan's original
sketch in his letter to Hamilton and a simple four colour map.

In 1878 Arthur Cayley (1821-1895) at a meeting of the London
Mathematical Society asked whether anyone had found a solution for
De Morgan's original question, but although there had been some
interest, no one had made any significant progress. Cayley became
interested in the problem and in 1879 published a short paper
On the colouring of maps
where he explained some of the difficulties in attempting a proof
and made some important contributions to the way the problem was
approached. His question that, 'if a particular map is already
successfully coloured with four colours, and we add another area,
can we still keep the same colouring?' began another line of
enquiry which led to the application of mathematical
induction to the problem.

Arthur Cayley (1821-1895)

Arthur Cayley showed that if four
colours had already been used to colour a map, and a new region was
added, it was not always possible to keep the original
colouring.

Above, all four colours have been used on the original map, and a
new region is drawn to surround it. In this case, a red region is
changed to blue, so that red can be used on the new surrounding
region.

Cayley also observed that it was possible to solve a version of
the problem by restricting the way the boundaries met. For example,
maps where just three countries met have three edges meeting at a
vertex. These are called 'cubic maps', and the maps used in the
following discussion are all cubic maps. Also, if a map can be
coloured with four colours, only three colours will appear on the
border.

The Patch demonstration. Imagine that
at some place in a map a number of countries meet at a point. Now
put a patch over the meeting point, and all the new meeting points
will have three borders emanating from them. These are cubic maps,
and a fourth colour can be used for the central region. On removing
the patch, we can return to the original colouring.

2. Some old techniques, new conditions and more problems!

In order to follow the developments of the problem, we need to
investigate briefly some of the ideas, procedures and techniques
that mathematicians developed in their attempts to solve it.

The Only Five Neighbours Conjecture [see note 1 below ]

'If you can't solve a
particular problem, find an easier one that you can solve.'
(Polya. How to Solve It
)

'Every map has at least one
country with five or fewer neighbours.'

Imagine a map of an island surrounded by the sea. In the
colouring of the countries of the island, we count the sea as one
region. Some countries may have only two borders (a digon), some
three (as in a triangle), some four (a square) and some five (a
pentagon) or more.

The simplest possible
configurations for surrounding a central region.
Note that in all of these configurations each node has only three
edges.

In 1813, Euler's formula for polyhedra was
adapted for two dimensions by Augustin Cauchy (1759-1857) by
projecting the polyhedron onto a plane thus forming the net of the
solid. In this way the formula became $f + v - e = 1$, because
Cauchy did not count the 'outside' region of the net.

Augustin Cauchy (1759-1857)

Imagine squashing the red cube down
onto a plane so that its base is opened out to form the outside
edge of the green net. Cauchy's idea was to cut out one face of the
cube, so that for a plane polygon, $f + v - e = 1$. Alternatively,
if the 'outside' of the net is regarded as a face with infinite
area, then we still have $f + v - e = 2$

We can assume there are at least three border lines (edges)
emerging from each meeting point (vertices).

So, from Euler's formula we get $3v$ edges in total. But, each
edge has a vertex at the other end, and might be counted twice, so
the total number of edges is at
least $\frac{3}{2} v$. So $e \geq\frac{3}{2}v$ or, $v \leq
\frac{2}{3}e$.

The proof that the map has at least one country surrounded by
five or fewer neighbours proceeds by contradiction
. If this leads to an absurdity, we have a proof.

Assume there is a map where every country ($f$) is surrounded by
at least six neighbours. If we select one country, and count all
the border lines ($e$) of the countries surrounding that one, we
have six borders. This will occur for all the countries in the map,
so the total number of borders will be $6f$ (where $f$ is the total
number of countries in the map). However, each border line has been
counted twice, so we need $\frac{6}{2}f$ which means $e \geq3f$ or
$f \leq\frac{1}{3}e$.

Now put these values into Euler's formula: $$f + v - e = 2$$ and
we have $$1/3(e) + 2/3(e) - e$$ which is zero!

This is the absurdity, so our original assumption was false.
This means that there must be at
least one country with five or fewer neighbours!

Minimal Criminals!

Another way to tackle the four colour problem is to assume it is
false, and see where this leads. Suppose there are maps that need
five colours or more, and we pick the maps with the smallest
possible number of countries. These maps are called minimal counter-examples or
minimal criminals !

So this means that a minimal criminal cannot be coloured with four colours,
but any map with fewer countries can be coloured with four colours. If
we can show that minimal criminals cannot exist, then we might be able to
make some progress.

For example, we can show that a minimal criminal cannot contain
a digon.

From the original map, take away a boundary from the digon, and
we get a new map with fewer countries. This map can be coloured
with four colours (from our assumption). We then colour this new
map, (we only need two colours). Now replace the border we removed,
and re-colour the map. We have used three colours, and since there
is still one more colour available, this shows that our map can be
coloured with four colours. But this is against our assumption,
so a minimal criminal cannot
contain a digon.

To show that a Minimal Criminal
cannot contain a region with two edges (a digon). Suppose there is
a minimal criminal that contains a digon. Removing an edge means
the map contains fewer regions. So this new map can be coloured
with four colours. Now replace the lost edge. Since only two
colours were needed before, replacing the edge means we can use a
third colour, and still have a fourth colour to use. So, a minimal
criminal can be coloured with four colours. Hence a minimal
criminal cannot contain a digon.

This procedure can be repeated to show that a minimal criminal
cannot contain a three-sided country (a trigon), but it breaks down
when we try the technique on a square, because when we replace the
square, the countries next to it may well be using all four
colours, so the proof procedure fails. Once this has happened, it
becomes obvious it won't work for pentagons, and so on.

The Six Colour Theorem

A similar technique can be applied to show that the six colour
theorem is true. First, we assume there are no maps that can be
coloured with six colours. Some of the maps can be coloured with
seven colours, so selecting one of these (a minimal criminal), if
we can show that it is possible to colour it with less than seven
colours, we have achieved our aim.

From the proof of the five neighbours theorem, it is possible to
proceed using the minimal criminal idea to show that any map can be coloured with six
colours!

3. From Regions to Knots, Networks and Topology

In 1879 Alfred Kempe (1849-1922), using techniques similar to
those described above, started from the 'five neighbours property'
and developed a procedure known as the method of 'Kempe Chains' to
find a proof of the Four Colour Theorem. He published this proof in
the American Journal of Mathematics. He found two simpler versions
that were published in the next year, and his proof stood for ten
years before Percy Heawood (1861-1955) showed there was an
important error in the proof-method that Kempe had used.

In 1880 P.G. Tait (1831-1901) a mathematical physicist, offered
a solution to the problem. Independently, Tait had established that
maps where an even number of boundary lines meet at every point,
could be coloured with two colours, although this result had
appeared earlier in Kempe's papers.

During 1876-77 Tait became well-known for his study and
classification of knots. At that time there were a number of
different theories about the structure of atoms. William Thompson
(Later, Lord Kelvin 1824-1907) inspired by the experiments of the
German Physicist Hermann von Helmholtz (1821-1894) proposed a
theory that atoms were knotted tubes of ether. Kelvin's theory of
'vortex atoms' was taken seriously for about twenty years, and it
inspired Tait to undertake a classification of knots. Tait, Thomson
and James Clark Maxwell (1831-1879) invented many topological ideas
during their studies. However, Kelvin's theory was fundamentally
mistaken and physicists lost interest in Tait's work.[see note 2 below ]

Tait began with the ways in which a
single closed loop of cord could be knotted. He had no systematic
method at the start, and began in an intuitive way by taking a
single closed loop and experimenting with the ways in which it
could be knotted. Of course, the cord had to be open (like a
shoelace) then knotted, and joined. Note that if you follow the
cord around the knot, the 'over - under' crossings will alternate.
He then went on to experiment with two loops and the ways in which
they could be knotted together. Shown here are knots with up to six
crossings for a single loop.

One of the outcomes of Tait's study was his Hamiltonian graph conjecture.

A map is regarded as a polyhedron drawn on a sphere, and it can
then be projected onto a plane. Tait proposed that any cubic
polyhedral map has a Hamiltonian
cycle [see note 3
below ]. Tait's method focused on the edges of the graph and
he showed that a Hamiltonian cycle could produce a four-coloring of
a map. It was not until 1946 that William Tutte (1917-2002) found
the first counterexample to Tait's conjecture.

Tait and the connection with knots

Tait
initiated the study of snarks in 1880, when he proved that the four
colour theorem was equivalent to the statement that no snark is
planar. A planar graph is one that can be drawn in the plane with
no edges crossing. It looks as if Tait's idea of non-planar graphs
might have come from his study of knots and Hamiltonian paths .

The first known snark was the Petersen graph discovered in 1898,
and mathematicians began to hunt for more of these kinds of graphs
but it was not until 1946 that another snark was found.

Snarks are projections of three dimensional graphs onto the
plane. There are no vertices where the blue edges appear to cross
each other. Snarks have the following properties:

The graph is cubic - three edges meet at every vertex.

The graph is bridgeless - it is impossible to cut the graph
into two separate pieces by deleting one edge.

Using three colours, there is no proper edge colouring for the
graph. All the edges meeting at a vertex cannot be coloured with
three different colours.

The edges meeting the vertices of this snark are coloured blue,
green and brown, but we always reach a stage where this process
cannot be continued.

Julius Peterson (1839-1910)

The Hunting of the
Snark is a poem written by Lewis Carroll, and Martin Gardner
named these graphs Snarks, because they were so elusive.

4. Transforming the problem and finding new methods.

Although Heawood found the major flaw in Kempe's proof method in
1890, he was unable to go on to prove the four colour theorem, but
he made a significant breakthrough and proved conclusively that
all maps could be coloured
withfivecolours.

Heawood made many important contributions to the problem,
shifting the focus of attention from the areas of a map, to the
borders between them. By 1898 he had proved that if the number of
edges around each region is divisible by 3 then the regions could
be coloured with four colours.

Cauchy's proof of Euler's formula also included the idea that
any net of a polyhedron can be triangulated by adding edges to make
non-triangular faces into triangles. He then developed a procedure
whereby he deleted the edges one by one, showing that Euler's
formula could be maintained at each step.[see note 4 below ]

Cauchy's Proof of Euler's Formula

Cauchy's 1813 proof of Euler's Formula began with the idea of a
projection of a polyhedron to obtain a plane net. He further
demonstrated (a) that any net could be triangulated, and his proof
(b) of Euler's Formula was accepted at the time.

(a)

In principle, every polygonal net can be triangulated. In this
net of a cube (a), $f + v - e$ is $10 + 8 - 17 = 1$, and Euler's
formula still holds.

(b)

Cauchy's argument was to remove the external edges from diagram
(a) one by one, and when he reached a stage as in diagram (b)
removed the whole triangle T, thus preserving Euler's formula. Many
mathematicians of the early nineteenth century agreed that this
procedure demonstrated a proof of Euler's formula for all
polyhedra.

By 1900, mathematicians knew that a planar graph can be
constructed from any map using the powerful concept of duality [see note 5 below ]. In the dual, the
regions are represented by vertices and two vertices are joined by
an edge if the regions are adjacent. In these graphs, the Four
Colour Conjecture now asks if the vertices of the graph can be
coloured with 4 colours so that no two adjacent vertices are the
same colour.

The 3-coloured map on the left has
$8$ regions $10$ vertices and $17$ edges. Its dual graph on the
right has $9$ regions $9$ vertices and $17$ edges where the
vertices are coloured the same as the areas of the map. The green
vertex at the bottom of the graph represents the infinite external
area for the map. Both the original map and its dual obey Euler's
Formula for networks $f + v - e = 1$ or, $\text{regions} +
\text{vertices} - \text{edges} = 1$. The duality relation is
symmetric: the dual of the dual will be the original graph, where
regions and vertices are exchanged.

During the first half of the twentieth century, mathematicians
focused on modifying these kinds of techniques to reduce
complicated maps to special cases which could be identified and
classified, to investigate their particular properties and
developed the idea of a minimal set of map configurations that
could be tested.

In the first instance, the set was thought to contain nearly
9,000 members which was an enormous task, and so the mathematicians
turned to computer techniques to write algorithms that could do the
testing for them. The algorithms used modified versions of Kempe's
original idea of chains together with other techniques to reduce
the number of members of the minimal set.

After collaborating with John Koch on the problem of
reducibility, in 1976 at the University of Illinois, Kenneth Appel
and Wolfgang Haken eventually reduced the testing problem to an
unavoidable set with 1,936 configurations, and a complete solution
to the Four Colour Conjecture was achieved. This problem of
checking the reducibility of the maps one by one was double checked
with different programs and different computers. Their proof showed
that at least one map with the smallest possible number of regions
requiring five colours cannot exist.

Since the first proof, more efficient algorithms have been found
for 4-coloring maps and by 1994, the unavoidable set of
configurations had been reduced to 633.

Is a 'Proof' Done on a Computer a Proper Proof?

Because the proof was done with the aid of a computer, there was
an immediate outcry. Many mathematicians and philosophers claimed
that the proof was not legitimate. Some said that proofs should
only be 'proved' by people, not machines, while others, of a more
practical mind questioned the reliability of both the algorithms
and the ability of the machines to carry them out without error.
However, many of the proofs written by mathematicians have been
found to be faulty, so the argument about reliability seems empty.
Whatever the opinions expressed, the situation produced a serious
discussion about the nature of proof which still continues
today.

For pedagogical notes:

Notes

More detail of this and the other procedures found in this
section can be seen in Robin Wilson's book Four Colours Suffice .

Knots can be left-handed or right-handed, and today there are
important applications of this property in chemistry, pharmacy,
biology and physics. (See Pedagogical Notes)

Named after William Rowan Hamilton (1805-1865). A Hamiltonian path in a graph visits
each vertex exactly once. A Hamiltonian cycle (or circuit) is a
path which visits each vertex exactly once and returns to the
starting vertex.(See Pedagogical Notes)

The book by Imre Lakatos, Proofs and Refutations has a
discussion and criticism of Cauchy's procedure (pages 6 - 12), and
much more on the story of Euler's Theorem.

The idea of duality arose in the 16th and 17th centuries with
developments in projective geometry. Mathematicians like Pascal and
Desargues found that new theorems could be found by exchanging the
terms 'point' and 'line' in descriptions of certain geometrical
configurations. An example is in regular polyhedra, where the
vertices of one correspond to the faces of the other. So the dual
of a tetrahedron is another tetrahedron, and the dual of a cube is
an octahedron. The dual of the dual is the original
polyhedron.

References

The very best popular, easy to read book on the Four Colour
Theorem is:
Wilson, R. (2003)Four Colours Suffice
.
London. Penguin Books.

This one brings us up to date, with more recent foundations and
philosophy.
Fritsch, R and Fritsch, G (2000)The Four Color Theorem: History,
Topological Foundations, and Idea of Proof
New York. Springer-Verlag.

Katz, V (1998)

A History of Mathematics; An
Introduction .

New York. Harlow, England. Addison Wesley Longman

Hardly any general history book has much on the subject, but the
last chapter in Katz called 'Computers and Applications' has a
section on Graph Theory, and the Four Colour Theorem is mentioned
twice.

Polya G. How to Solve
It.
This is the classic book about Problem Solving. There have been
many editions of this book since it first appeared in the 1950s and
it is still easily available. Curiously, recent editions have been
given the subtitle 'A new aspect of Mathematical Method'.

Lakatos, I. (1976) Proofs and
Refutations: The Logic of Mathematical Discovery.
Cambridge. C.U.P.
This is another important book which led to the research into
Problem Solving and Investigations in the 1970s. It begins as a
classroom discussion between a teacher and a group of students
about the proof of Euler's formula, and ranges through the ideas,
objections and possibilities that were actually discussed by
mathematicians and scientists in the nineteenth century. It raises
some of the most important issues about teaching and learning
problem solving and about mathematical methods and proof.

Related References

I have had a little book on String Games for some time. When I
was at school it was called Cat's Cradle, and we played it in our
break time.

Recently, a French journal has published a paper on the
'algebra' of string figures! If you go to Amazon you will find a
nice book by Ann Swain and Michael Taylor called Finger Strings: A Book of Cats Cradles and
String Figures to be published by Floris books in September
2008. There are some 80 figures described with coloured diagrams.
It's spiral bound, so it will stay open while you follow the
instructions. It also comes with a couple of string loops!

For knot experts, The Ashley
Book of Knots is a classic for anyone interested in the
hundreds of different kinds of knots and their uses. Amazon has
various editions available at different prices.

For Graph Theory, Wikipedia gives a good overview, and you can
skip the really technical stuff. It shows the kinds of modern
applications of this area of mathematics. If you go to Graph
Colouring and click on the Four Colour Theorem, then you will find
a lot more information.http://en.wikipedia.org/wiki/Graph_theory